Biomedical Engineering Reference
In-Depth Information
When the internal (dispersed) fluid is very viscous—like concentrated alginate
solutions—its flow rate is much smaller than that of the external (continuous) fluid.
Under this assumption, (4.106) can be simplified, yielding
*
*
q
=
A p
(
+
B
-
1)
(4.107)
where
A R
=
(
+
2
R f R
)
e
s
i
(4.108)
B R R
=
(
+
2
R f
)
e
e
s
The coefficients
A
and
B
are deduced from the expressions of the hydraulic resis-
tances
R
e
,
R
i
, and
R
s
,
which in term are computed by the Washburn law adapted for
rectangular channels [87]. At this stage, two important remarks should be made:
all the parameters in (4.107) and (4.108) are known except the alginate viscosity
h
i
and the two-phase additional pressure drop coefficient
f
.
First, the flow velocity—hence the shear rate—of the alginate solution in the
feeding channel is very small. In consequence, a zero shear alginate viscosity
h
i
0
can
be used for the determination of the hydraulic resistance of the alginate flow in the
feeding channel.
Second, the value of the function
f
is determined by remarking that the droplets
have a size approximately equal to that of the cross section of the channel, and are
separated by relatively large distances. In such a case, one can expect excess pres-
sure drop due to the droplets to be small [39, 40]. We can then assume
f
~1, which
is experimentally verified. An important consequence of
f
being close to unity is
that it can be taken as a constant; then the coefficient
B
in (4.107) is purely a geo-
metrical factor (the viscosity
h
e
cancels out), while
A
is proportional to
h
e
/
h
i
and
D
is proportional to
h
i
/
h
e
.
The model is confirmed by experimental observations. Figure 4.72 shows the
different flow regimes in the [
Q
i
,
Q
e
] coordinates system together with a compari-
son between model and experimental results.
Figure 4.72
Diagram of flow regimes for alginate (1.25 wt%)/oil binary system with flow-rate ac-
tuation (same legends as Figure 4.71).
